The Defense Department began working yesterday with a private marketing firm to create a database of high school students ages 16 to 18 and all college students to help the military identify potential recruits in a time of dwindling enlistment in some branches. The program is provoking a furor among privacy advocates. The new database will include personal information including birth dates, Social Security numbers, e-mail addresses, grade-point averages, ethnicity and what subjects the students are studying.